Stochastic Modeling of Groundwater Flow by Unconditional and Conditional Probabilities: The Inverse Problem

The inverse problem is defined here as follows: determine the transmissivity at varius points, given the shape and boundary of the aquifer and recharge intensity and given a set of measured log‐transmissivity Y and head H values at a few points. The log‐transmissivity distribution is regarded as a realization of a random function of normal and stationary unconditional probability density function (pdf). The solution of the inverse problem is the conditional normal pdf of Y , conditioned on measured H and Y , which is expressed in terms of the unconditional joint pdf of Y and H . The problem is reduced to determining the unconditional head‐log‐transmissivity covariance and head variogram for a selected Y covariance which depends on a few unknown parameters. This is achieved by solving a first‐order approximation of the flow equations. The method is illustrated for an exponential Y covariance, and the effect of head and transmissivity measurements upon the reduction of uncertainty of Y is investigated systematically. It is shown that measurement of H has a lesser impact than those of Y , but a judicious combination may lead to significant reduction of the predicted variance of Y . Possible applications to real aquifers are outlined.